problem stringlengths 15 4.7k | solution stringlengths 2 11.9k | answer stringclasses 51
values | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
Consider the sequence $ a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.$
$ (a)$ Prove that $ 1<a_n$ and $ a_{n\plus{}1}<a_n$ for all $ n$.
$ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit.
$ (c)$ Determine $ \displaystyle\lim_{n\t... | 1. **Prove that \(1 < a_n < 3\) for all positive integers \(n\)**:
- **Base Case**: For \(n = 1\), we have \(a_1 = \frac{3}{2}\). Clearly, \(1 < \frac{3}{2} < 3\).
- **Inductive Step**: Assume \(1 < a_n < 3\) for some \(n\). We need to show that \(1 < a_{n+1} < 3\).
- **Prove \(a_{n+1} > 1\)**:
\[
... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties:
$a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer.
Determine all possible values of $c$ and the associated values of $a$ and $b$. | 1. Given a right triangle with perpendicular sides \(a\) and \(b\) and hypotenuse \(c\), we have the following properties:
\[
a = p^m \quad \text{and} \quad b = q^n
\]
where \(p\) and \(q\) are prime numbers, \(m\) and \(n\) are positive integers, and \(c = 2k + 1\) with \(k\) a positive integer.
2. Accord... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$? | 1. **Define the problem and the sums:**
We are given five distinct real numbers \(a_1, a_2, a_3, a_4, a_5\). We need to consider all possible sums of the form \(a_i + a_j\) where \(i \neq j\). There are \(\binom{5}{2} = 10\) such sums since each pair \((i, j)\) is considered once.
2. **Calculate the number of disti... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$1+2+3+4+5+6=6+7+8$.
What is the smallest number $k$ greater than $6$ for which:
$1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ? | 1. We start with the given equation:
\[
1 + 2 + 3 + \cdots + k = k + (k+1) + (k+2) + \cdots + n
\]
We need to find the smallest number \( k \) greater than 6 for which this equation holds, with \( n \) being an integer greater than \( k \).
2. First, we use the formula for the sum of the first \( k \) natu... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Kira has $3$ blocks with the letter $A$, $3$ blocks with the letter $B$, and $3$ blocks with the letter $C$. She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the letter A have d... | To solve this problem, we need to determine the maximum number of distinct distances between blocks with the same letter in a sequence of 9 blocks, where each letter (A, B, C) appears exactly 3 times.
1. **Understanding the Problem:**
- We have 9 blocks in total: 3 A's, 3 B's, and 3 C's.
- We need to find the ma... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $m, n$, and $p$ are three different natural numbers, each between $2$ and $9$, what then are all the possible integer value(s) of the expression $\frac{m+n+p}{m+n}$? | 1. We start with the given expression $\frac{m+n+p}{m+n}$, where $m, n$, and $p$ are distinct natural numbers between $2$ and $9$.
2. First, we note that $m$ and $n$ are distinct natural numbers between $2$ and $9$. Therefore, the minimum value of $m+n$ is $2+3=5$ (since $m$ and $n$ are distinct), and the maximum valu... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks.
How many cows will eat all the grass on six he... | 1. Let \( x \) be the amount of hectares that a cow eats per week, and \( y \) be the amount of hectares of grass that grows per week.
2. From the problem, we have two scenarios:
- In two weeks, three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers. | 1. **Understanding the problem**: We need to find the smallest positive value of \(36^k - 5^m\) where \(k\) and \(m\) are positive integers.
2. **Modulo 5 analysis**:
- Note that \(36 \equiv 1 \pmod{5}\). Therefore, \(36^k \equiv 1^k \equiv 1 \pmod{5}\).
- This implies \(36^k - 5^m \equiv 1 - 0 \equiv 1 \pmod{5... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $ b$, $c$, $d$ are non-negative integers.
| 1. We start by considering the expression $\frac{2^a - 2^b}{2^c - 2^d}$, where $a, b, c, d$ are non-negative integers. Without loss of generality, we can assume $a > b$ and $c > d$.
2. For the expression to be an integer, the denominator $2^c - 2^d$ must divide the numerator $2^a - 2^b$. We can factor both the numerat... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$2019$ circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours? | 1. **Assigning Numbers to Regions**:
For each region \(i\), assign a number \(a_i\) such that all points in region \(i\) are contained inside exactly \(a_i\) of the 2019 circles. This number \(a_i\) represents the count of circles that enclose the region \(i\).
2. **Parity Argument**:
We claim that no region \(... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each of the $10$ dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice crea... | 1. Let's define the variables as follows:
- \( v_T \): Number of dwarfs who like vanilla ice cream and tell the truth.
- \( v_F \): Number of dwarfs who like vanilla ice cream and lie.
- \( c_T \): Number of dwarfs who like chocolate ice cream and tell the truth.
- \( c_F \): Number of dwarfs who like choco... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$? | 1. **Assume $n \geq 5$ and derive a contradiction:**
- Let $\{x_1, x_2, x_3, \ldots, x_n\}$ be a set of $n$ distinct positive integers such that the sum of any three of them is a prime number.
- Consider the residues of these integers modulo 3. There are three possible residues: 0, 1, and 2.
2. **Pigeonhole Prin... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of the smallest and largest possible values for $x$ which satisfy the following equation.
$$9^{x+1} + 2187 = 3^{6x-x^2}.$$ | 1. Start with the given equation:
\[
9^{x+1} + 2187 = 3^{6x - x^2}
\]
2. Rewrite \(9^{x+1}\) and \(2187\) in terms of base 3:
\[
9^{x+1} = (3^2)^{x+1} = 3^{2(x+1)} = 3^{2x+2}
\]
\[
2187 = 3^7
\]
3. Substitute these into the original equation:
\[
3^{2x+2} + 3^7 = 3^{6x - x^2}
\]
4.... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime). | 1. **Consider the problem statement**: We need to find all primes \( p \) that can be written both as a sum and as a difference of two primes.
2. **Set up the equations**: Let \( p \) be a prime such that:
\[
p = x + y \quad \text{and} \quad p = z - w
\]
where \( x, y, z, w \) are primes.
3. **Consider t... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals $5$. What is the greatest n... | 1. **Understanding the Problem:**
We are given a $4 \times 4$ grid where each cell contains a non-negative integer. We need to ensure that the sum of any five cells that can be covered by the given figures equals 5. The figures can be moved and rotated.
2. **Analyzing the Figures:**
The figures are not explicitl... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x,y$ and $z$ be positive real numbers such that $xy+z^2=8$. Determine the smallest possible value of the expression $$\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.$$ | 1. Given the constraint \(xy + z^2 = 8\), we need to find the minimum value of the expression:
\[
\frac{x+y}{z} + \frac{y+z}{x^2} + \frac{z+x}{y^2}.
\]
2. To approach this problem, we can use the method of Lagrange multipliers or apply inequalities such as the AM-GM inequality. Here, we will use the AM-GM ine... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$, let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true:
$1)$ For any two friends $X'$... | 1. **Graph Representation**:
- Consider a graph \( G \) where each vertex represents a contestant.
- There is an edge between two vertices \( A \) and \( B \) if and only if the contestants \( A \) and \( B \) are friends.
- Each vertex \( A \) is assigned a value \( t(A) \), which is the total score that cont... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $12$ dentists in a clinic near a school. The students of the $5$th year, who are $29$, attend the clinic. Each dentist serves at least $2$ students. Determine the greater number of students that can attend to a single dentist . | 1. We start by noting that there are 12 dentists and 29 students. Each dentist serves at least 2 students. Therefore, the minimum number of students served by all dentists is:
\[
12 \times 2 = 24 \text{ students}
\]
2. Since there are 29 students in total, the number of students left after each dentist serves ... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each number from the set $\{1, 2, 3, 4, 5, 6, 7\}$ must be written in each circle of the diagram, so that the sum of any three [i]aligned[/i] numbers is the same (e.g., $A+D+E = D+C+B$). What number cannot be placed on the circle $E$? | 1. **Define the problem and variables:**
We are given the set $\{1, 2, 3, 4, 5, 6, 7\}$ and need to place each number in a circle such that the sum of any three aligned numbers is the same. Let this common sum be denoted by $s$.
2. **Set up the equations:**
Consider the three vertical columns:
\[
(A + D + ... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$? | 1. Given that \( p = \frac{b}{4} \sqrt{\frac{2a - b}{2a + b}} \) is a prime number, we need to find the maximum possible value of \( p \).
2. First, we note that if \( b \) is odd, then \( p \) is not an integer. Therefore, \( b \) must be even. Let \( b = 2c \) for some positive integer \( c \).
3. Substituting \( b =... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all positive integers $n$ such that $ xy+1 \equiv 0 \; \pmod{n} $ implies that $ x+y \equiv 0 \; \pmod{n}$. | To determine all positive integers \( n \) such that \( xy + 1 \equiv 0 \pmod{n} \) implies \( x + y \equiv 0 \pmod{n} \), we proceed as follows:
1. **Assume \( xy + 1 \equiv 0 \pmod{n} \)**:
\[
xy \equiv -1 \pmod{n}
\]
2. **Consider \( x \) and \( y \) such that \( x \) is coprime to \( n \)**:
Since \( ... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list] | 1. **Show that \( n \) is a quadratic residue \(\pmod{p}\):**
Given \( p = 4n + 1 \), we need to show that \( n \) is a quadratic residue modulo \( p \).
Since \( p \equiv 1 \pmod{4} \), we know that \( -1 \) is a quadratic residue modulo \( p \). This means there exists an integer \( i \) such that \( i^2 \equ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_{1}={11}^{11}$, $a_{2}={12}^{12}$, $a_{3}={13}^{13}$, and \[a_{n}= \vert a_{n-1}-a_{n-2}\vert+\vert a_{n-2}-a_{n-3}\vert, n \ge 4.\] Determine $a_{{14}^{14}}$. | 1. We start by defining the sequence \(a_n\) as given:
\[
a_1 = 11^{11}, \quad a_2 = 12^{12}, \quad a_3 = 13^{13}
\]
and for \(n \geq 4\),
\[
a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|
\]
2. Define the difference \(\Delta a_n = a_{n+1} - a_n\). Then, we can express the recurrence relation in ... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence? | To determine the maximal number of successive odd terms in a sequence where each term is obtained by adding its largest digit to the previous term, we need to analyze the behavior of the sequence based on the last digit of each term.
1. **Initial Considerations**:
- If a term ends in 9, the next term will be even b... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest integer $n \ge 4$ for which one can choose four different numbers $a, b, c, $ and $d$ from any $n$ distinct integers such that $a+b-c-d$ is divisible by $20$ . | To determine the smallest integer \( n \ge 4 \) for which one can choose four different numbers \( a, b, c, \) and \( d \) from any \( n \) distinct integers such that \( a + b - c - d \) is divisible by \( 20 \), we can use the Pigeonhole Principle (PHP).
1. **Understanding the Problem:**
We need to find the small... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Bob play the following number-guessing game. Alice writes down a list of positive integers $x_{1}$, $\cdots$, $x_{n}$, but does not reveal them to Bob, who will try to determine the numbers by asking Alice questions. Bob chooses a list of positive integers $a_{1}$, $\cdots$, $a_{n}$ and asks Alice to tell him... | **
One round does not suffice because if Bob only asks for one sum, there could be multiple sets of \( x_i \) that produce the same sum. For example, if Alice's numbers are such that the result can be represented in more than one way as an integral linear combination of the given \( a_k \), Bob would not be able to ... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The set $M= \{1;2;3;\ldots ; 29;30\}$ is divided in $k$
subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an
integer number $)$, then $a$ and $b$ belong different subsets.
Determine the minimum value of $k$. | 1. **Identify the pairs \((a, b)\) such that \(a + b = n^2\):**
We need to find all pairs \((a, b)\) in the set \(M = \{1, 2, 3, \ldots, 30\}\) such that \(a + b = n^2\) for some integer \(n\). The possible values of \(n^2\) within the range of \(M\) are \(4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, ... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
All the squares of a board of $(n+1)\times(n-1)$ squares
are painted with [b]three colors[/b] such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of $n$. | To solve the problem, we need to find the greatest possible value of \( n \) such that a \((n+1) \times (n-1)\) board can be painted with three colors in a way that no four squares forming a rectangle have the same color.
1. **Understanding the Problem:**
- We need to ensure that for any two different columns and ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest term of the sequence $a_1, a_2, a_3, \ldots$ defined by $a_1=2014^{2015^{2016}}$ and
$$
a_{n+1}=
\begin{cases}
\frac{a_n}{2} & \text{ if } a_n \text{ is even} \\
a_n + 7 & \text{ if } a_n \text{ is odd} \\
\end{cases}
$$ | 1. **Initial Term Analysis**:
The sequence starts with \( a_1 = 2014^{2015^{2016}} \). We need to determine the smallest term in the sequence defined by the recurrence relation:
\[
a_{n+1} =
\begin{cases}
\frac{a_n}{2} & \text{if } a_n \text{ is even} \\
a_n + 7 & \text{if } a_n \text{ is odd}
\en... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a table $4\times 4$ we put $k$ blocks such that
i) Each block covers exactly 2 cells
ii) Each cell is covered by, at least, one block
iii) If we delete a block; there is, at least, one cell that is not covered.
Find the maximum value of $k$.
Note: The blocks can overlap. | To solve this problem, we need to find the maximum number of blocks \( k \) that can be placed on a \( 4 \times 4 \) grid such that:
1. Each block covers exactly 2 cells.
2. Each cell is covered by at least one block.
3. If we delete any block, there is at least one cell that is not covered.
Let's break down the solut... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the numbers from $1$ to $32$. A game is made by placing all the numbers in pairs and replacing each pair with the largest prime divisor of the sum of the numbers of that couple. For example, if we match the $32$ numbers as: $(1, 2), (3,4),(5, 6), (7, 8),..., (27, 28),(29, 30), (31,32)$, we get the following l... | 1. **Initial Pairing and Sum Calculation:**
We start by pairing the numbers from 1 to 32. Let's consider the pairs as follows:
\[
(1, 32), (2, 31), (3, 30), \ldots, (16, 17)
\]
The sums of these pairs are:
\[
33, 33, 33, \ldots, 33 \quad \text{(16 times)}
\]
2. **Finding the Largest Prime Divis... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$ | 1. Let \( P(x) \) be the assertion \( f(f(x)) = \frac{x^2 - x}{2} f(x) + 2 - x \).
2. Consider \( P(2) \):
\[
f(f(2)) = \frac{2^2 - 2}{2} f(2) + 2 - 2 = f(2)
\]
This implies that \( f(2) \) is a fixed point of \( f \), i.e., \( f(f(2)) = f(2) \).
3. Suppose \( f(u) = u \) for some \( u \). Then, substitut... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Every single point on the plane with integer coordinates is coloured either red, green or blue. Find the least possible positive integer $n$ with the following property: no matter how the points are coloured, there is always a triangle with area $n$ that has its $3$ vertices with the same colour.
| 1. **Understanding the Problem:**
We need to find the smallest positive integer \( n \) such that no matter how we color the points on the plane with integer coordinates using three colors (red, green, blue), there will always be a triangle with area \( n \) whose vertices are all the same color.
2. **Area of a Tri... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n> 2$ be an integer. A child has $n^2$ candies, which are distributed in $n$ boxes. An operation consists in choosing two boxes that together contain an even number of candies and redistribute the candy from those boxes so that both contain the same amount of candy. Determine all the values of $n$ for which the ch... | 1. **Define the Problem and Notations:**
- Let \( n > 2 \) be an integer.
- A child has \( n^2 \) candies distributed in \( n \) boxes.
- An operation consists of choosing two boxes that together contain an even number of candies and redistributing the candies so that both boxes contain the same amount.
- W... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remainin... | 1. **Understanding the Problem:**
- We need to cover a $6 \times 6$ board using $18$ dominoes.
- Each domino is a $1 \times 2$ or $2 \times 1$ rectangle.
- We need to find the smallest positive integer $k$ such that placing $k$ dominoes on the board leaves a unique way to cover the remaining part of the board ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board?
Clarification: Two squares are [i]neighbors [/i] if they have a common side. | To solve this problem, we need to ensure that every \(3 \times 3\) subboard on a \(7 \times 8\) board has at least two neighboring black squares. We aim to find the minimum number of black squares required to satisfy this condition.
1. **Initial Setup and Constraints**:
- The board is \(7 \times 8\), which means it... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$. | To find all the real numbers \( k \) that satisfy the given property, we need to ensure that for any non-zero real numbers \( a \) and \( b \), at least one of the numbers \( a \), \( b \), or \( \frac{5}{a^2} + \frac{6}{b^3} \) is less than or equal to \( k \).
1. **Consider the case when \( a = b = 2 \):**
\[
... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form $ x^3\plus{}2y^2$, with $ x, y$ being integers. | To determine the maximal possible length of the sequence of consecutive integers which are expressible in the form \( x^3 + 2y^2 \), where \( x \) and \( y \) are integers, we need to analyze the possible values of \( x^3 + 2y^2 \) modulo 8.
1. **Analyze \( x^3 \mod 8 \):**
- If \( x \equiv 0 \pmod{8} \), then \( x... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer. | 1. **Define the problem constraints:**
- We need to fill an $11 \times 11$ board with numbers $-1$, $0$, or $1$.
- The sum of the numbers in each column must be nonnegative.
- The sum of the numbers in each row must be nonpositive.
2. **Analyze the sum conditions:**
- Let $C_i$ be the sum of the $i$-th col... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$. The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.
[img]https:... | 1. **Identify the coordinates of the vertices:**
- Let the cube be positioned in a 3D coordinate system with \( A = (0, 0, 0) \), \( B = (2, 0, 0) \), \( C = (2, 2, 0) \), \( D = (0, 2, 0) \), \( A' = (0, 0, 2) \), \( B' = (2, 0, 2) \), \( C' = (2, 2, 2) \), and \( D' = (0, 2, 2) \).
- The point \( K \) is the mi... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$. | 1. We start with the given equation:
\[
3n = 999^{1000}
\]
2. To find the unity digit of \( n \), we first determine the unity digit of \( 999^{1000} \). Notice that:
\[
999 \equiv -1 \pmod{10}
\]
3. Therefore:
\[
999^{1000} \equiv (-1)^{1000} \pmod{10}
\]
4. Since \( (-1)^{1000} = 1 \):
... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In each square of the table below, we must write a different integer from $1$ to $17$, such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from $1$ to $17$ can be omitted?
[img]https://wiki... | 1. **Sum of integers from 1 to 17**:
The sum of the first \( n \) positive integers is given by the formula:
\[
S = \frac{n(n+1)}{2}
\]
For \( n = 17 \):
\[
S = \frac{17 \times 18}{2} = 153
\]
2. **Sum of the numbers in the top and bottom rows**:
Let the sum of the numbers in the bottom row ... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and eve... | To solve this problem, we need to determine the minimum number of concerts required so that each musician has the opportunity to listen to every other musician at least once. This problem can be approached using combinatorial design theory, specifically the concept of strongly separating families.
1. **Understanding t... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Twelve balls are numbered by the numbers $1,2,3,\cdots,12$. Each ball is colored either red or green, so that the following two conditions are satisfied:
(i) If two balls marked by different numbers $a$ and $b$ are colored red and $a+b<13$, then the ball marked by the number $a+b$ is colored red, too.
(ii) If two bal... | 1. **Assume without loss of generality (WLOG) that ball 1 is colored red.** We will consider the cases where ball 1 is red and then multiply the total number of cases by 2 to account for the symmetry of exchanging red with green.
2. **Identify the minimum red-colored ball besides 1.** Let \( a \) be the smallest numbe... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a cycling competition with $14$ stages, one each day, and $100$ participants, a competitor was characterized by finishing $93^{\text{rd}}$ each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.) | 1. **Understanding the problem**: We need to determine the best possible overall standing for a competitor who finishes 93rd in each of the 14 stages of a cycling competition with 100 participants. The overall standings are based on the total cycling time over all stages.
2. **Analyzing the given information**: The co... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a game, there are several tiles of different colors and scores. Two white tiles are equal to three yellow tiles, a yellow tile equals $5$ red chips, $3$ red tile are equal to $ 8$ black tiles, and a black tile is worth $15$.
i) Find the values of all the tiles.
ii) Determine in how many ways the tiles can be chose... | ### Part I: Values of all types of tiles
1. **Determine the value of a black tile:**
\[
\text{A black tile} = 15 \text{ chips}
\]
2. **Determine the value of a red tile:**
Given that 3 red tiles are equal to 8 black tiles:
\[
3 \text{ red tiles} = 8 \text{ black tiles}
\]
Since each black tile... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have? | 1. **Define the arithmetic progression and its properties:**
Let the first term of the arithmetic progression (AP) be \( a \) and the common difference be \( d = 2 \). Suppose the AP has \( k \) terms. The terms of the AP are:
\[
a, a+2, a+4, \ldots, a+2(k-1)
\]
2. **Sum of the terms of the AP:**
The su... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property? | 1. **Understanding the Problem:**
- We have a large square divided into \(25\) unit squares, forming a \(5 \times 5\) grid.
- We need to draw diagonals in such a way that no two diagonals share any points.
2. **Analyzing the Grid:**
- Each unit square can have two possible diagonals: one from the top-left to ... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation
$$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$
has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $
[b]b)[/b] Find the floor of $ m=\sqrt{2+\s... | ### Part (a)
1. **Given:**
\[
a, b \text{ are non-negative integers such that } a^2 > b.
\]
We need to show that the equation
\[
\left\lfloor \sqrt{x^2 + 2ax + b} \right\rfloor = x + a - 1
\]
has an infinite number of solutions in the non-negative integers.
2. **Consider the expression inside... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies
$$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$
Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $ | 1. **Given Inequality Analysis**:
We start with the given inequality for the sequence \( (x_n)_{n \ge 0} \):
\[
x_n x_m + k_1 k_2 \le k_1 x_n + k_2 x_m, \quad \forall m, n \in \{0\} \cup \mathbb{N}.
\]
Let's consider the case when \( m = n \):
\[
x_n^2 + k_1 k_2 \le k_1 x_n + k_2 x_n.
\]
Simp... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$.
[i]Daniel Jinga[/i] | 1. Given \( A \in \mathcal{M}_2(\mathbb{R}) \) such that \(\det(A) = d \neq 0\) and \(\det(A + dA^*) = 0\), we need to prove that \(\det(A - dA^*) = 4\).
2. Let \(\text{tr}(A) = t\). The adjugate of \(A\), denoted \(\text{adj}(A)\), satisfies the relation \(A \cdot \text{adj}(A) = \det(A) \cdot I = d \cdot I\). For a ... | 4 | Algebra | proof | Yes | Yes | aops_forum | false |
[b]a)[/b] Show that the number $ \sqrt{9-\sqrt{77}}\cdot\sqrt {2}\cdot\left(\sqrt{11}-\sqrt{7}\right)\cdot\left( 9+\sqrt{77}\right) $ is natural.
[b]b)[/b] Consider two real numbers $ x,y $ such that $ xy=6 $ and $ x,y>2. $ Show that $ x+y<5. $ | ### Part A:
We need to show that the number \( N = \sqrt{9 - \sqrt{77}} \cdot \sqrt{2} \cdot \left( \sqrt{11} - \sqrt{7} \right) \cdot \left( 9 + \sqrt{77} \right) \) is a natural number.
1. First, simplify the expression inside the square roots:
\[
N = \sqrt{9 - \sqrt{77}} \cdot \sqrt{2} \cdot \left( \sqrt{11} ... | 8 | Inequalities | proof | Yes | Yes | aops_forum | false |
[b]a)[/b] Show that if two non-negative integers $ p,q $ satisfy the property that both $ \sqrt{2p-q} $ and $ \sqrt{2p+q} $ are non-negative integers, then $ q $ is even.
[b]b)[/b] Determine how many natural numbers $ m $ are there such that $ \sqrt{2m-4030} $ and $ \sqrt{2m+4030} $ are both natural. | ### Part A:
1. Given that \( \sqrt{2p - q} \) and \( \sqrt{2p + q} \) are non-negative integers, let:
\[
\sqrt{2p - q} = n \quad \text{and} \quad \sqrt{2p + q} = m
\]
where \( n \) and \( m \) are non-negative integers.
2. Squaring both sides, we get:
\[
2p - q = n^2 \quad \text{and} \quad 2p + q = m... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence
$$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$
is convergent and calculate its limit. | 1. **Define the sequence and the function:**
Let \( f: [0,1] \to [0,1] \) be a nondecreasing function. We need to prove that the sequence
\[
\left( \int_0^1 \frac{1+f^n(x)}{1+f^{n+1}(x)} \, dx \right)_{n \ge 1}
\]
is convergent and calculate its limit.
2. **Pointwise behavior of the integrand:**
Con... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
[b]a)[/b] Show that the expression $ x^3-5x^2+8x-4 $ is nonegative, for every $ x\in [1,\infty ) . $
[b]b)[/b] Determine $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $ | **Part (a):**
1. We start with the expression \( x^3 - 5x^2 + 8x - 4 \).
2. We need to show that this expression is non-negative for every \( x \in [1, \infty) \).
3. Let's factorize the expression:
\[
x^3 - 5x^2 + 8x - 4 = (x-1)(x-2)^2
\]
4. To verify the factorization, we can expand the right-hand side:
... | 8 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $
[b]a)[/b] Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $
[b]b)[/b] Find the biggest real number $ a $ for which... | Given the sequence \( \left( a_n \right)_{n \ge 1} \) of real numbers such that \( a_1 > 2 \) and \( a_{n+1} = a_1 + \frac{2}{a_n} \) for all natural numbers \( n \), we need to show the following:
**a)** Show that \( a_{2n-1} + a_{2n} > 4 \) for all natural numbers \( n \), and \( \lim_{n \to \infty} a_n = 2 \).
1. ... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b\in\mathbb{R},~a>1,~b>0.$ Find the least possible value for $\alpha$ such that :$$(a+b)^x\geq a^x+b,~(\forall)x\geq\alpha.$$ | 1. We need to find the least possible value for $\alpha$ such that the inequality
\[
(a+b)^x \geq a^x + b
\]
holds for all $x \geq \alpha$, given that $a > 1$ and $b > 0$.
2. Let's start by testing $\alpha = 1$. We need to check if the inequality holds for $x = 1$:
\[
(a+b)^1 \geq a^1 + b
\]
S... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Consider two distinct positive integers $a$ and $b$ having integer arithmetic, geometric and harmonic means. Find the minimal value of $|a-b|$.
[i]Mircea Fianu[/i] | To find the minimal value of \( |a - b| \) for two distinct positive integers \( a \) and \( b \) such that their arithmetic mean, geometric mean, and harmonic mean are all integers, we proceed as follows:
1. **Arithmetic Mean Condition**:
\[
A = \frac{a + b}{2} \in \mathbb{N} \implies a + b \text{ is even} \imp... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximum possible real value of the number $ k$, such that
\[ (a \plus{} b \plus{} c)\left (\frac {1}{a \plus{} b} \plus{} \frac {1}{c \plus{} b} \plus{} \frac {1}{a \plus{} c} \minus{} k \right )\ge k\]
for all real numbers $ a,b,c\ge 0$ with $ a \plus{} b \plus{} c \equal{} ab \plus{} bc \plus{} ca$. | To determine the maximum possible real value of \( k \) such that
\[
(a + b + c) \left( \frac{1}{a + b} + \frac{1}{b + c} + \frac{1}{c + a} - k \right) \ge k
\]
for all real numbers \( a, b, c \ge 0 \) with \( a + b + c = ab + bc + ca \), we proceed as follows:
1. **Substitute specific values to find an upper bound fo... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions:
$ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$.
Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$.
| 1. Given the conditions \( ac = bd \) and \( \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \), we start by analyzing the second condition.
2. Notice that the equality \( \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \) suggests a symmetry. We can use the AM-GM inequality to gain insight:
\[... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest positive integer $n$ such that the following is true:
There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$. | 1. **Understanding the problem**: We need to find the largest positive integer \( n \) such that there exist \( n \) distinct positive integers \( x_1, x_2, \ldots, x_n \) with the property that for any choice of \( a_1, a_2, \ldots, a_n \in \{-1, 0, 1\} \), not all zero, the sum \( \sum_{k=1}^n a_k x_k \) is not divis... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers.
a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty.
b) Show that if $G_p\cap G_q$ is a ... | ### Part (a)
1. Consider the functions \( f_p(x) = \frac{1}{p}x + p \) and \( f_q(x) = \frac{1}{q}x + q \).
2. To find the intersection \( G_p \cap G_q \), we need to solve \( f_p(x) = f_q(x) \):
\[
\frac{1}{p}x + p = \frac{1}{q}x + q
\]
3. Rearrange the equation to isolate \( x \):
\[
\frac{1}{p}x - \fr... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Prove that the equation $x^2+y^2+z^2+t^2=2^{2004}$, where $0 \leq x \leq y \leq z \leq t$, has exactly $2$ solutions in $\mathbb Z$.
[i]Mihai Baluna[/i] | 1. **Initial Setup and Transformation:**
Given the equation \( x^2 + y^2 + z^2 + t^2 = 2^{2004} \) with \( 0 \leq x \leq y \leq z \leq t \), we introduce the variables \( x_n = \frac{x}{2^n}, y_n = \frac{y}{2^n}, z_n = \frac{z}{2^n}, t_n = \frac{t}{2^n} \). This transforms the equation into:
\[
x_n^2 + y_n^2 +... | 2 | Number Theory | proof | Yes | Yes | aops_forum | false |
Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value.
[i]Mircea Becheanu[/i] | 1. **Understanding the problem**: We need to find the maximum determinant of a \(3 \times 3\) matrix \(A\) where each row and each column contains the distinct real numbers \(a, b, c\). Additionally, we need to determine the number of such matrices that achieve this maximum determinant value.
2. **Matrix structure**: ... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivale... | **
- If any \( x_i = 0 \), then all \( x_i = 0 \) for \( i = 1, 2, \ldots, n \). This is because if \( P(x_i, x_{i+1}) = 0 \) and \( x_i = 0 \), then \( x_{i+1} \) must also be 0 to satisfy the equation \( P(0, x_{i+1}) = x_{i+1}^2 = 0 \). Thus, the trivial solution is \( x_i = 0 \) for all \( i \).
2. **Non-trivia... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection. | 1. **Restate the problem in graph theory terms:**
We need to find the smallest \( n \) such that a complete graph \( K_{11} \) can be covered by some \( K_n \) subgraphs, where each vertex is included in at most 5 of these \( K_n \) subgraphs.
2. **Consider the case \( n = 3 \):**
- If we cover \( K_{11} \) with... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. Find the number of polynomials $P(x)$ with coefficients in $\{0, 1, 2, 3\}$ for which $P(2) = n$. | To find the number of polynomials \( P(x) \) with coefficients in \(\{0, 1, 2, 3\}\) such that \( P(2) = n \), we need to express \( n \) in base 4. This is because the coefficients of the polynomial \( P(x) \) are restricted to the set \(\{0, 1, 2, 3\}\), which corresponds to the digits in base 4.
1. **Express \( n \... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and
\[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \] | 1. **Clarify the sequence definition:**
The sequence is defined as follows:
\[
a_{n+1} = \begin{cases}
\frac{a_n}{2} & \text{if } a_n \text{ is even}, \\
a_n + 7 & \text{if } a_n \text{ is odd}.
\end{cases}
\]
The initial term is \( a_1 = 1993^{1994^{1995}} \).
2. **Determine the parity of \( a... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $ n$,define $ f(n)\equal{}lcm(1,2,...,n)$.
(a)Prove that for every $ k$ there exist $ k$ consecutive positive integers on which $ f$ is constant.
(b)Find the maximum possible cardinality of a set of consecutive positive integers on which $ f$ is strictly increasing and find all sets for whic... | ### Part (a)
1. **Define the function \( f(n) \):**
\[
f(n) = \text{lcm}(1, 2, \ldots, n)
\]
where \(\text{lcm}\) denotes the least common multiple.
2. **Observation:**
For any integer \( n \), \( f(n) \) is the least common multiple of the first \( n \) positive integers. If we consider \( f(n) \) and ... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest positive integer $n$ for which there exist $n$ nonnegative integers $x_1, x_2,\ldots , x_n$, not all zero, such that for any $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ from the set $\{-1, 0, 1\}$, not all zero, $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$ is not divi... | 1. **Assume \( n \geq 10 \)**:
- There are \( 3^n - 1 \) non-zero expressions of the form \( \sum_{i=1}^n \varepsilon_i x_i \) with \( \varepsilon_i \in \{-1, 0, 1\} \).
- Since \( 3^n - 1 > n^3 \) for \( n \geq 10 \), by the Pigeonhole Principle, at least two of these expressions must have the same remainder mod... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least $n\in N$ such that among any $n$ rays in space sharing a common origin there exist two which form an acute angle. | 1. **Base Case:**
- Consider the case when \( d = 1 \). In one-dimensional space, any two non-null vectors (rays) will either point in the same direction or in opposite directions. If they point in opposite directions, they form an angle of \( 180^\circ \), which is not acute. Therefore, the largest number of vector... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.
[i]Dinu Șerbănescu[/i] | 1. **Identify the given angles and the cyclic nature of the pentagon:**
- Given angles: $\angle B = 120^\circ$, $\angle C = 120^\circ$, $\angle D = 130^\circ$, $\angle E = 100^\circ$.
- Since $ABCDE$ is a cyclic pentagon, it is inscribed in a circle with center $O$.
2. **Use the known property of cyclic polygons... | 1 | Geometry | proof | Yes | Yes | aops_forum | false |
Find the least number $ c$ satisfyng the condition $\sum_{i=1}^n {x_i}^2\leq cn$
and all real numbers $x_1,x_2,...,x_n$ are greater than or equal to $-1$ such that $\sum_{i=1}^n {x_i}^3=0$ | To find the least number \( c \) satisfying the condition \(\sum_{i=1}^n {x_i}^2 \leq cn\) for all real numbers \(x_1, x_2, \ldots, x_n \geq -1\) such that \(\sum_{i=1}^n {x_i}^3 = 0\), we can proceed as follows:
1. **Define the Functions and Constraints:**
- Define \( g: \mathbb{R}^n \rightarrow \mathbb{R} \) by \... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Given an integer $n \geq 2$ determine the integral part of the number
$ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}}) \dots ({1+\frac {k} {n})}}$ - $\sum_{k=1}^{n-1} (1-\frac {1} {n}) \dots(1-\frac{k}{n})$ | To solve the problem, we need to determine the integral part of the expression:
\[
\sum_{k=1}^{n-1} \frac{1}{(1+\frac{1}{n}) \cdots (1+\frac{k}{n})} - \sum_{k=1}^{n-1} (1-\frac{1}{n}) \cdots (1-\frac{k}{n})
\]
We will denote the two sums separately and analyze them.
1. **Denote the first sum:**
\[
S_1 = \sum_{... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sequence of integers $ \left( a_n\right)_{n\ge 0} $ defined as
$$ a_n=\left\{\begin{matrix}n^6-2017, & 7|n\\ \frac{1}{7}\left( n^6-2017\right) , & 7\not | n\end{matrix}\right. . $$
Determine the largest length a string of consecutive terms from this sequence sharing a common divisor greater than $
1 $ may... | 1. **Define the sequence and the problem:**
The sequence of integers \( (a_n)_{n \ge 0} \) is defined as:
\[
a_n = \begin{cases}
n^6 - 2017, & \text{if } 7 \mid n \\
\frac{1}{7}(n^6 - 2017), & \text{if } 7 \nmid n
\end{cases}
\]
We need to determine the largest length of a string of consecutiv... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\] | 1. Consider the polynomial \( P(X) = \prod_{i=1}^n (X^2 + X a_i + b_i) \). This polynomial has degree \(2n\) and has \(2n\) roots, namely \(a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n\).
2. Since \(P(X)\) has \(2n\) roots, we can write it as \( P(X) = \prod_{i=1}^n (X - a_i)(X - b_i) \).
3. Suppose \(n \geq 3\). We wi... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve? | 1. **Igor's Strategy:**
- Igor can ensure that the final polynomial has at most 2 distinct integer roots by setting \(a_0 = 1\) in the first move. This is because the integer roots of the polynomial must be divisors of the constant term \(a_0\). Since \(a_0 = 1\), the possible integer roots are \(\pm 1\).
2. **Egor... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors? | 1. **Understanding the Problem:**
- We have a $100 \times 100$ board.
- Each cell is divided into two triangles by drawing a diagonal.
- We need to paint these triangles such that any two triangles sharing a common side or vertex have different colors.
- We need to find the smallest number of colors require... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Point $P$ and equilateral triangle $ABC$ satisfy $|AP|=2$, $|BP|=3$. Maximize $|CP|$. | 1. **Apply Ptolemy's Inequality**: For a cyclic quadrilateral \(ACBP\), Ptolemy's Inequality states:
\[
AP \cdot BC + BP \cdot AC \geq CP \cdot AB
\]
Since \(ABC\) is an equilateral triangle, we have \(AB = BC = CA\). Let the side length of the equilateral triangle be \(s\).
2. **Substitute the given lengt... | 5 | Geometry | other | Yes | Yes | aops_forum | false |
Given $1962$ -digit number. It is divisible by $9$. Let $x$ be the sum of its digits. Let the sum of the digits of $x$ be $y$. Let the sum of the digits of $y$ be $z$. Find $z$. | 1. Given a $1962$-digit number that is divisible by $9$, we need to find the value of $z$, where $z$ is the sum of the digits of the sum of the digits of the sum of the digits of the original number.
2. Let $x$ be the sum of the digits of the original number. Since the number is divisible by $9$, $x$ must also be divis... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given is a number with $1998$ digits which is divisible by $9$. Let $x$ be the sum of its digits, let $y$ be the sum of the digits of $x$, and $z$ the sum of the digits of $y$. Find $z$. | 1. **Understanding the problem**: We are given a number \( n \) with 1998 digits that is divisible by 9. We need to find the sum of the digits of \( n \), then the sum of the digits of that sum, and finally the sum of the digits of that result. We denote these sums as \( x \), \( y \), and \( z \) respectively.
2. **S... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Among all the numbers representable as $36^k - 5^l$ ($k$ and $l$ are natural numbers) find the smallest.
Prove that it is really the smallest. | 1. **Understanding the problem**: We need to find the smallest positive value of the expression \(36^k - 5^\ell\) where \(k\) and \(\ell\) are natural numbers.
2. **Analyzing modulo 10**:
- For any natural number \(k\), \(36^k \equiv 6^k \equiv 6 \pmod{10}\).
- For any natural number \(\ell\), \(5^\ell \equiv ... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
a) Find the minimal value of the polynomial $P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$
b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$. | a) To find the minimal value of the polynomial \( P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2 \), we can rewrite it in a more convenient form:
\[ P(x,y) = 4 + x^2y^2(x^2y^2 + x^2 + y^2 - 3). \]
Let's analyze the expression \( x^2y^2 + x^2 + y^2 - 3 \).
1. **Case 1: \( x^2 + y^2 \geq 3 \)**
\[
x^2y^2 + x^2 + y^2 - 3 ... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Every member, starting from the third one, of two sequences $\{a_n\}$ and $\{b_n\}$ equals to the sum of two preceding ones. First members are: $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$. How many natural numbers are encountered in both sequences (may be on the different places)? | 1. We start by defining the sequences $\{a_n\}$ and $\{b_n\}$ based on the given initial conditions and the recurrence relation. Specifically, we have:
\[
a_1 = 1, \quad a_2 = 2, \quad a_n = a_{n-1} + a_{n-2} \quad \text{for} \quad n \geq 3
\]
\[
b_1 = 2, \quad b_2 = 1, \quad b_n = b_{n-1} + b_{n-2} \qua... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares? | 1. **Initial Assumption and Coloring:**
- Suppose we do not need any $1 \times 1$ tiles.
- Color the $23 \times 23$ square in a checkerboard pattern, starting with the first row as black. This means alternate rows are black and white.
- In this pattern, there will be $12$ black rows and $11$ white rows, each c... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots,... | To solve this problem, we need to analyze the strategies of both players and determine the maximum number of quadratic equations that the first player can ensure do not have real roots, regardless of the second player's actions.
1. **Understanding the Quadratic Equation**:
A quadratic equation is of the form \( ax^... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). ... | 1. **Define the Problem and Variables:**
- Let the three boxes be \( A \), \( B \), and \( C \).
- Let the initial number of stones in boxes \( A \), \( B \), and \( C \) be \( a \), \( b \), and \( c \) respectively.
- Sisyphus moves stones between these boxes, and Zeus rewards or charges him based on the dif... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$. Such an arrangement is called [i]successful[/i] if each number is the product of its neighbors. Find the number of successful arrangements. | 1. **Reformulate the Problem:**
Each square of a \((2^n-1) \times (2^n-1)\) board contains either \(1\) or \(-1\). An arrangement is called *successful* if each number is the product of its neighbors. We need to find the number of successful arrangements.
2. **Transform the Problem to \(\mathbb{F}_2\):**
For a s... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$. | 1. Let \( A \) be a natural number whose decimal digits form an increasing sequence. We can represent \( A \) as \( A = \overline{a_n a_{n-1} a_{n-2} \dots a_0}_{10} \), where \( a_n, a_{n-1}, \dots, a_0 \) are the digits of \( A \) and \( a_n < a_{n-1} < \cdots < a_0 \).
2. To find the sum of the digits of \( 9A \), ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We are given five equal-looking weights of pairwise distinct masses. For any three weights $A$, $B$, $C$, we can check by a measuring if $m(A) < m(B) < m(C)$, where $m(X)$ denotes the mass of a weight $X$ (the answer is [i]yes[/i] or [i]no[/i].) Can we always arrange the masses of the weights in the increasing order ... | 1. **Label the weights**: Let the weights be labeled as \( A, B, C, D, E \).
2. **Determine the order of three weights**: We need to determine the order of \( A, B, \) and \( C \). There are \( 3! = 6 \) possible permutations of these three weights. We can determine the correct order using at most 5 comparisons:
- ... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$. | 1. **Define the set \( M \) and the condition:**
Let \( M \) be a finite set of numbers such that among any three of its elements, there are two whose sum belongs to \( M \).
2. **Establish the upper bound:**
We need to show that \( M \) can have at most 7 elements. To see this, consider the set \( M = \{-3, -2,... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and column... | 1. **Horizontal Direction Analysis:**
- Label the columns from $1$ to $n$. If a vector goes from column $i$ to $j$, we assign it the integer value $j-i$.
- Consider the sum of all these values taken over all vectors in the magic square.
2. **Vectors with Tail at Column $k$:**
- Let the numbers in column $k$ b... | 0 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$. Heracle defeats a hydra by cutting it into two parts which ... | 1. **Restate the problem in graph theory terms:**
- Each head of the hydra is a vertex.
- Each neck of the hydra is an edge connecting two vertices.
- When a vertex \( v \) is hit, all edges incident to \( v \) are removed, and new edges are added to connect \( v \) to all vertices it was not previously connec... | 10 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to $ 100$. How many cells are there that are on the distance $ 50$ from each of the three cells? | 1. **Define the problem and set up coordinates:**
We are given three cells on an infinite chessboard with pairwise distances of 100. We need to find the number of cells that are at a distance of 50 from each of these three cells. We start by placing one of the cells at the origin, \((0,0)\).
2. **Characterize the s... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime. | To find all natural numbers \( x \) for which \( 3x+1 \) and \( 6x-2 \) are perfect squares, and the number \( 6x^2-1 \) is prime, we proceed as follows:
1. **Express \( 3x+1 \) and \( 6x-2 \) as perfect squares:**
\[
3x + 1 = k^2 \quad \text{and} \quad 6x - 2 = t^2
\]
where \( k \) and \( t \) are integer... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the coefficients of $x^{17}$ and $x^{18}$ after expansion and collecting the terms of $(1+x^5+x^7)^{20}$. | To find the coefficients of \(x^{17}\) and \(x^{18}\) in the expansion of \((1 + x^5 + x^7)^{20}\), we need to consider the multinomial expansion and the possible combinations of the exponents that sum to 17 and 18.
1. **Identify the possible combinations for \(x^{17}\):**
- We need to find the combinations of \(0,... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img] | 1. **Define the Geometry and Coordinates:**
Given a square \(ABCD\) with vertices \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\). Two parallel lines intersect the square, and the distance between these lines is \(1\).
2. **Identify the Points of Intersection:**
Let's denote the points of intersection of the ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Nine numbers $a, b, c, \dots$ are arranged around a circle. All numbers of the form $a+b^c, \dots$ are prime. What is the largest possible number of different numbers among $a, b, c, \dots$? | 1. **Understanding the Problem:**
We are given nine numbers \(a, b, c, \dots\) arranged in a circle such that all expressions of the form \(a + b^c\) are prime. We need to determine the largest possible number of different numbers among \(a, b, c, \dots\).
2. **Initial Assumptions:**
Let's assume the numbers are... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On each non-boundary unit segment of an $8\times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers? | 1. **Define the problem and notation:**
- Let \( N \) be the total number of ways to tile an \( 8 \times 8 \) chessboard with dominoes.
- Consider each non-boundary unit segment on the chessboard. There are \( 56 \) horizontal and \( 56 \) vertical non-boundary unit segments, making a total of \( 112 \) such segm... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called [i]good[/i] if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$. Denote by $\mathcal{J}$ the set of polynomials of the form ... | 1. **Understanding the Problem:**
We need to find the smallest natural number \( D \) such that every monomial of degree \( D \) can be expressed as a sum of products of good polynomials and other polynomials with integer coefficients.
2. **Properties of Good Polynomials:**
A polynomial \( f(x_1, \ldots, x_n) \)... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.